On the Construction of Generalised Bobillier Formula

Tulay Erisir; Mehmet Ali Gungor; Soley Ersoy

Home/Archive/arXiv

arXiv · 1503.01616

On the Construction of Generalised Bobillier Formula

Tulay Erisir·Mehmet Ali Gungor·Soley Ersoy

View original source ↗Read source document ↗Export metadata ↓

Abstract

In this study, we consider the generalized complex number system $C_{\rm{p}} = \left\{ {x + iy\;:\;x,y \in R,\;{i^2} = {\rm{p}} \in R} \right\}$ corresponding to elliptical complex number, parabolic complex number and hyperbolic complex number systems for the special cases of ${\rm{p}} < 0,\;{\rm{p}} = 0,\;{\rm{p}} > 0$, respectively. This system is used to derive Bobillier Formula in the generalized complex plane. In accordance with this purpose we obtain this formula by two different methods for one-parameter planar motion in ${C_{\rm{p}}}$; the first method depends on using the geometrical interpretation of the generalized Euler-Savary formula and the second one uses the usual relations of the velocities and accelerations.

Explore related subjects

math.GM

Keep this discovery

BibTeX RIS

Tulay Erisir, Mehmet Ali Gungor, Soley Ersoy. 2015-03-05. On the Construction of Generalised Bobillier Formula. https://arxiv.org/abs/1503.01616

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Average Chord Lengths in a Triangle

Let $P$ be a point inside a triangle $T$. We consider the average length of the chords of $T$ through $P$, where the direction of the chord is chosen uniformly. An elementary formula is obtained in terms of the distances from $P$ to the sides and vertices of the triangle. Several classical triangle centers give especially simple specializations. For example, if $I$ is the incenter, then \[ M_T(I)=\frac{2r}{\pi} \log\left(\cot\frac A4\cot\frac B4\cot\frac C4\right). \] Our main result is the sharp inequality \[ M_T(P)\le \frac{p}{\pi\sqrt3}\log(2+\sqrt3), \] valid simultaneously for every triangle of perimeter $p$ and every interior point $P$. Thus, among all such pairs $(T,P)$, the largest possible average chord length occurs only when $T$ is equilateral and $P$ is its center. The proof is an elementary symmetrization argument. We close with brief remarks relating the problem to the radial center of a convex body, the electrostatic potential center of a triangle, and dual quermassintegrals.

math.GM↗

arXiv2026

The decrease in sectional curvature of the Fisher--Rao manifold of beta distributions

In the paper, the author develops a general convexity lemma and applies it to demonstrate the monotonic decrease of a function involving polygamma functions, arising from the sectional curvature of the Fisher--Rao manifold of beta distributions.

math.GM↗

arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. We prove a conjecture of Liu stating that \[\sum_{\mathrm{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \geq 2+\left(\frac{2r}{R}\right)^k,~~k>1, \] with the reverse inequality for $0<k<1$. The proof reduces the problem to three positive variables with fixed sum and product. We also determine the equality cases.

math.GM↗